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PD.4 Tangent Planes and Linear Approximations

One of the most important ideas in single variable calculus is that as we zoom in toward a point on the graph of a differentiable function, the graph becomes indistinguishable for its tangent line and we can approximate the function by a linear function. In this section we develop similar ideas in three dimensions. As we zoom in toward a point on a surface that is the graph of a differentiable function of two variables, the surface looks more and more like a plane (its tangent plane) and we can approximate the function by a linear function of two variables. We also extend the idea of a differential to functions of two or more variables.

 

Tangent Planes

A tangent plane is defined to be a plane that is just tangent to a surface (it touches the surface in only one point). The definition is given below:

 

Suppose $f$ has a continuous partial derivatives. An equations of the tangent plane to the surface $z=f(x)$ at the point MATH is:


MATH

 

 

This is a picture of a tangent plane:



Linear Approximations

Linear Approximations are found by first finding the tangent plane to the surface. When the numbers that were used to form the equation are plugged back into the tangent plane, the tangent plane equation will return a value that is the same as the original surface. Now, if one were to put numbers that are close to the numbers that were used to form the tangent plane back into the tangent plane equation, one will get a value that is close to that of the actual surface. This is the linear approximation, which is in fact just the tangent plane. The equation is shown below:


MATH

 

 

This animation shows how a tangent plane resembles the actual surface as you get closer to the point of intersection:


 

 

Consider a function of two variables, $z=f(x,y)$, and suppose x changes from $a$ to $a+\Delta x$ and y changes from $y$ to $b+\Delta y.$Then the corresponding increment of z is:


MATH

 

Thus the in increment $\Delta z$ represents the change in the value of $f$ when $(x,y)$ changes from $(a,b)$ to MATH. Therefore we can define differentiability of a function of two variables as follows:

If $z=f(x,y),$then $f$ is differentiable at $(a,b)$ if $\Delta z$ can be expressed in the form


MATH

where $\varepsilon _{1}$ and MATH as MATH

 

It is sometimes hard to use the above definition directly to check the differentiability of a function, but the following theorem provides a convenient sufficient condition for differentiability:


If the partial derivatives $f_{x}$ and $f_{y}$ exist near $(a,b)$ and are continuous at $(a,b)$, then $f$ is differentiable at $(a,b).$
 

 

 

Differentials

For a function of two variables, $z=f(x,y),$ we define the differentials $dx$ and $dy$ to be independent variables; that is, they can be given any values Then the differential $dz$, also called the total differential, is defined by:


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Functions of Three or More Variables

Functions of three or more variables are similar to two variable functions:

For a linear approximation, simply use the following (which one will notice is similar to the two variable counterpart):


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The increment is defined as:


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The differential $dw$ is defined in terms of the differentials $dx,~dy,~dz$ of the independent variables by:


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